[ 370 ] quodque folum, certa nitri figna pr^ebcre, fed plura concurrere debere^ ut de vero nitro produdo dubium non relinquatur. LI I. AnEffay towards folvlng a Problem in the DoSirine of Chances. By the late Rev. Mr. Bayes, F. R. S. communicated by Mr. Price, in a Letter to John Canton, A. M. F. R. S. Dear Sir, Read Dec. 23, TT Now fend you an eflay which I have ^763- J^ found among the papers of our de- ceafed friend Mr. Bayes, and which, in my opinion, has great merit, and well deferves to be preferved. Experimental philofophy, you will find, is nearly in- terefted in the fubjed of it ; and on this account there feems to be particular reafon for thinking that a com- munication of it to the Royal Society cannot be im- proper. He had, you know, the honour of being a mem- ber of that illuftrious Society, and was much efteem- ed by many in it as a very able mathematician. In an introduaion which he has writ to this Eflay, he fays, that his defign at firft in thinking on the fubjea: of it was, to find out a method by which we might judge concerning the probability that an event has to hap- pen, in given circumftances, upon fuppofition that we know nothing concerning it but that, under the fame circum- [ 371 ] circumftances, it has happened a certain number of times, and failed a certain other number of times. He adds, that he foon perceived that it would not be very difficult to do this, provided fome rule could be found according to which we ought to eftimate the chance that the probability for the happening of an event perfectly unknown, fhould lie between any two named degrees of probability, antecedently to any ex- periments made about it ; and that it appeared to him that the rule muft be to fuppofe the chance the fame that it {hould lie between any two equidifferent de- grees ; which, if it were allowed, all the reft might be eafily calculated in the common method of pro- ceeding in the dodtrine of chances. Accordingly, I find among his papers a very ingenious folution of this problem in this way. But he afterwards confidered, that ihcpo/iulate on which he had argued might not perhaps be looked upon by all as reafonablej and therefore he chofe to lay down in another form the propofition in which he thought the folution of the problem is contained, and in ajcholium to fubjoin the reafons why he thought fo, rather than to take into his mathematical reafoning any thing that might ad- mit difpute. This, you will obferve, is the method which he has purfued in this efTay. Every judicious perfon will be fenfible that the problem now mentioned is by no means merely a curious fpeculation in the dodlrine of chances, but ne- ceflary to be folved in order to a fure foundation for all our reafonings concerning paft fa<3:s, and what is likely to be hereafter. Common fenfe is indeed fufficient to fhew us that, from the obfervation of what has in former inftances been the confequence of a certain 5 caufe [ 372 1 caufc or adion, one may make a judgment what is likely to be the confequence of it another time, and that the larger number of experiments we have to fupport a conclulion, fo much the more reafon v/e have to take it for granted. But it is certain that we cannot determine, at leaft not to any nicety, in what degree repeated experiments confirm a conclufion, without the particular difcuflion of the beforementi- oned problem ; which, therefore, is necefTary to be con- fidered by any one who would give a clear account of the ftrength of analogical or induSiive reafoniiig \ con- cerning, which at prefent, we feem to know little more than that it does fometimes in fad convince us, and at other times not ; and that, as it is the means of cquainting us with many truths, of which otherwife we muft have been ignorant ; fo it is, in all proba- bility, the fource of many errors, which perhaps might in fome meafure be avoided, if the force that this fort of reafoning ought to have with us were more diftin6tly and clearly underftood, Thefe obfervations prove that the problem enquired after in this eflay is no Icfs important than it is curi- ous. It may be fafely added, I fancy, that it is alfo a problem that has never before been folved* Mr. De Moivre, indeed, the great improver of this part of mathematics, has in his Law^ of chance *, after Ber- noulli, and to a greater degree of exadnefs, given rules to find the probability there is, that if a very great number of trials be made concerning any event, * See Mr. De Moivre's DoSfrine of Chances, p. 243, &c. H^ has omitted the demonftrations of his rules, but thefe have been fince fupplied by Mr, Simpfon at the conclufion of his treatife on The Nature and Laws of Chance*, the [ 373 ] the proportion of the number of times it will hap- pen, to the number of times it will fail in thofe tri- als, ihould differ lefs than by fmall affigned limits from the proportion of the probability of its happen- ing to the probability of its failing in one fingle trial. But I know of no perfon who has fhewn how to de- duce the folution of the converfe problem to this ; namely, « the number of times an unknown event has happened and .failed being given, to find the chance that the probability of its happening ihould lie fomewhere between any two named degrees of probability." What Mr. Dc Moivre has done therefore cannot be thought fufficient to make the confideration of this point unneceflkry : efpecially, as the rules he has given are not pretended to be rigo- roufly exadt, except on fuppofition that the number of trials made are infinite s from whence it is not ob- vious how large the number of trials muft be in or- der to make them exadt enough to be depended on in praftice, Mr. De Moivre calls the problem he has thus folv- ed, the hardeft that can be propofed on the fubjeft of chance. His folution he has applied to a very important purpofe, and thereby fhewn that thofe a remuch miftaken who have infinuated that the Doc- trine of Chances in mathematics is of trivial confe- quence, and cannot have a place in any ferious enqui- ry *'. The purpofe I mean is, to mew what reafon we have for believing that there are in the conftitutioh of things fixt laws according to which events happen, and that, therefore, the frame of the world muft be * See his DoSrine of Chances, p. 252, &c. Vol. LIIL Ccc the [ 374 ] the eflfed of the wifdom and power of an intelligent caufe; and thus to confirm the argument taken from final caufes for the exiftence of the Deity. It will be eafy to fee that the converfe problem folved in this effay is more diredly applicable to this purpofe -, for it ihews us, with diftinilnefs and precifion, in every cafe of any particular order or recurrency of events, what reafon there is to think that fuch recurrency or order is derived from ftable caufes or regulations inna-^ ture, and not from any of the irregularities of chance. The two laft rules in this eflfay are given without the dedudions of them. I have chofen to do this becaufe thefe deducStions, taking up a good deal of room, would fwell the e% too much ; and alfo be- caufe thefe rules, though of confiderable ufe, do not anfwer the purpofe for which they are given as per^ fedly as could be wiftied. They are however ready to be produced, if a communication of them fhould be thought proper, I have in fome places writ fhort notes, and to the whole I have added an application of the rules in the effay to fome particu- lar cafes, in order to convey a clearer idea of the na- ture of the problem, and to fhew how far the folu- tion of it has been carried. 1 am fenfible that your time is fo much taken up .hat I cannot r^afonably expeft that yoa fcould mi- nutely examine every part of what I now fend you* Some of the calculations, particularly in the Appen- dix, no one can make without a good deal of labour* I have taken fo much care about them, that I believe there can be no material error in any of them 5 but fliould there be any fuch errors, I am the only per- fon who ought to be confidered as anfwerable for them. Mr. [ 375 ] Mr. Bayes has thought fit to begin his work with a brief demonftration of the general laws of chance. His reafon for doing this, as he fays in his introduc- tion, was not merely that his reader might not ha?e the trouble of fearching elfewhere for the principles on which he has argued, but becaufe he did not know whither to refer him for a ckar demonftration of diem^ He has alfo made an apology for the peculiar definition he has given of the word chance or froba^ bilky. His defign herein was to cut qW all difpute about the meaning of the word^ which in common language is ufed in different fenfes by perfons of dif- ferent opinions, and according as it is applied to pafi ox future fa<5ts. But whatever different lenies it may have, all (he obferves) will allow that an exped:ation depending on the truth of any paji iz€k, or the hap- pening of zny future event, ought to be eftimated fo much the more valuable as the fad is more likely to be true, or the event more likely to happen. Inftead therefore, of die proper fenfe of the word frobabp- Utyy he has given that vi^hich all will allow to be its proper meafure in every cafe where the word is ufed. But it is time to conclude this letter. Experimental philofophy is indebted to you for feveral difcoveries and improvements i and, therefore, I cannot help thinking that there is a peculiar propriety in direft- ing to you the following effay and appendix. That your enquiries may be rewarded with many further fiiccefies, and that you may enjoy every eveiy valuable bleffing, is the fincere wiih of. Sir, your very humble fervant, Newington-Green, ^. ^ t ^ * Nov. 10, 1763. Richard Price* Ccc 2 SEC- [ 376 ] P R O B L E M. Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a lingle trial lies fomewhere between any two degrees of pro- bability that can be named. D SECTION L E F I N I T I O N I. Several events are in-^ conjtjienty when if one of them happens, none of the reft can. 2. Two events are contrary when one, or other of them mufti and both together cannot happen, 3. An event is faid to fail, when it cannot hap- pen s or, which comes to the fame thing, when its con- trary has happened. 4. An event is faid to be determined when it has either happened or failed. 5. The probability of any event is the ratio between the value at which an expedation depending on the happening of the event ought to be computed, and the value of the thing expeded upon it's happening. 6. By chance I mean the fame as probability. 7. Events are independent when the happening of any one of them does neither increafe nor abate the probability of the reft.. PROP. r. When feveral events are inconliftent the probabili-- ty of the happening of one or other of them is the fum of the probabilities of each of them. Suppofe [ 377 ] Suppofe there be three fuch events, and which ever of them happens I am to receive N, and that the pro- bability of the I ft, 2d, and 3d are refpedtively ^, l^> ~. Then (by the definition of probability) the va- lue of my expedation from the i ft will be a, from the 2d ^, and from the 3d ^. Wherefore the value of my expeftations from all three will be ^4-^ +- ^. But the fum of my expectations from all three is in this cafe an expeftation of receiving N upon the hap- pening of one or other of them. Wherefore (by de- finition 5) the probability of one or other of them is bilities of each of them. Corollary. If it be certain that one or other of the three events muft happen, then a -^ if -\'C •= N. For in this cafe all the expectations to- gether amounting to a certain expedation of re- ceiving N, their values together muft be equal to N, And from hence it is plain that the proba- bility of an event added to the probability of its fai- lure (or of its contrary) is the ratio of equality. For thefe are two inconfiftent events, one of which ne- ceiTarily happens. Wherefore if the probability of P , ^5 ^p an event is — that of it's failure will be ■ — rv - . PROP. 2. If a perfon has an expedation depending on the happening of an event, the probability of the event is to the probability of its failure as his lofs if it fails to his gain if it happens. Suppofe a perfon has an expedation of receiving N, depending on an event the probability of which is [378 ] • P is rr. Then (by definition 5) the value of his ex- pedation is P, and therefore if the event fail, he lofes that which in value is P ; and if it happens he re- ceives N, but his expedation ceafes. His gain there- fore is N — ^P. Likewife fince the probability of the event is —, that of its failure (by corollary prop, i) • N— P ^r^ P . N— P T> . , XT -D • IS - ^ ■■-, But rr- IS to -r-T— as F IS to JN — 1 , 1. e, N N N the probability of the event is to the probability of it's failure, as his lofs if it fails to his gain if it happens. PROP. 3. The probability that two fubfequent events will both happen is a ratio compounded of the probabi- lity of the I ft, and the probability of the 2d on fup- pofition the ift happens. Suppofe that, if both events happen, I am to receive N, that the probability both will happen is ^ , that the I ft will is ~ (and confequently that the ift will not is -~-) and that the ad will happen upon fup- pofition the ift does is rr-. Then (by definition 5) P will be the value of my expedation, which will be- come b if the ift happens. Confequently if the ift happens, my gain by it is ^ — P, and if it fails my lofs is P. Wherefore, by the foregoing propofition, ^ is to ^, i. e. a is to N— ^ as P is to ^—P. Where- fore (componendo inverfe) ^ is to N as P is to k But the ratio of P to N is compounded of the ratio of P to ^, and that of i to N. Wherefore the 5 f^^^ [ 379 ] fame ratio of P to N is compounded of the ratio of ^ to N and that of b to N, i. e, the probability that the two fubfequent events will both happen is com- pounded of the probability of the i ft and the proba- bility of the 2d on fuppolition the i ft happens. Corollary. Hence if of two fubfequent events the probability of the ift be ^, and the probability of both together be—, then the probability of the 2d on fuppolition the ift happens is -. PRO P. 4. If there be two fubfequent events to be determined every day, and each day the probability of the 2d is |j and the probability of both |^, and I am to re- ceive N if both the events happen the ift day on which the 2d does^l fay, according to thefe con- p ditions, the probability of my obtaining N is j. For if not, let the probability of my obtaining N be :|r and let ^ be to at as N— -iJ to N. Then fince ^ is the probability of my obtaining N (by definition i) x is the value of my expedation. And again, becaufe ac- cording to the foregoing conditions the ift day I have an expeftation of obtaining N depending on the hap- pening of both the events together, the probability of which is — , the value of this expectation is P. Like- wife, if this coincident fhould not happen I have an expedation of being reinftated in my former circum- ftances, i.e. of receiving that which in value is ^ de- pending t 380 ] pending on the failure of the 2d event the probability of which (by cor. prop, i) is -r^ or-, becaufe y is to X as N — b to N. Wherefore fince x is the thing expeded and - the probability of obtaining it, the value of this expedlation is y. But thefe two laft ex- pedations together are evidently the fame with my original expectation, the value of which is x, and therefore P ^y = x. But ^ is to at as N — b is to N. Wherefore x is to P as N is to by and ^ (the probability of my obtaining N) is 7. Cor. Suppofe after the expectation given me in the foregoing propofition, and before it is at all known whether the ift event has happened or not, I fliould find that the 2d event has happened; from hence I can only infer that the event is determined on which my expectation depended, and have no reafon to efteem the value of m.y expectation either greater or lefs than it was before. For if I have reafon to think it lefs, it would be reafbnable for me to give fomething to be reinftated in my former circumftances, and this over and over again as often as I fhould be in- formed that the 2d event had happened, which is evi- dently abfurd* And the like abfurdity plainly follows if you fay I ought to fet a greater value on my expec- tation than before, for then it would be reafonable for me to refufe fomething if offered me upon condition I would relinquifh it, and be reinftated in my former circumftances ; and this likewife over and over again as often as (nothing being known concerning the ifl event) it fhould appear that the 2d had happened. Notwithfl^nding therefore this difcovery that the 2d event [ 38i ] event has happened, my expedation ought to be efteemed the fame in value as before, i. e. ^, and confequently the probability of my obtaining X P N is (by definition 5) flill |;t or t-"*. But after this difcovery the probability of my obtaining N is the pro- bability that the ift of two fubfequent events has hap- pened upon the fuppoiition that the 2d has, whofe pro- babilities were as before fpecified. But the probability that an event has happened is the fame as the proba- bility I have to gueis right if I guefs it has happened. Wherefore the following propofition is evident. PRO P. 5: If there be two fubfequent events, the probability of the 2d i-a„d the probability of both togeAer \, and it being ift difcovered that the 2d event has hap- pened, from hence I guefs that the ift event has al- fo happened, the probability I am in the rightis t ''• PROP. * What is here faid may perhaps be a little illuftrated by con* fidering that all that can be loft by the happening of the 2d event is the chance I fhould have had of being reinftated in my former circumftances, if the event on which my expeftation depended had been determined in the manner exprefled in the propofition. But this chance is always as much againji me as it isj^r me. If the I ft event happens, it is againji me, and equal to the chance for the 2d event's failing* If the ift event does not happen, it is for me, and equal alfo to the chance for the 2d event's failing. The lofs of it, therefore, can be no difadvantage. f What is proved by Mr. Hayes in this and the preceding pro- pofition is the fame with the anfwer to the following queftion. What is the probability that a certain event, when it happens, will Vol. LIIL D d d be [382 ] P R O P. 6. The probability that feveral independent events ihall all happen is a ratio compounded of the proba- bilities of each. For from the nature of independent events, the probability that any one happens is not altered by the happening or failing of any of the reft, and confe- quently the probability that the ad event happens on luppofition the ift does is the fame with its original probability ; but the probability that any two events happen is a ratio compounded of the probability of the ift event, and the probability of the 2d on fuppofition the I ft happens by prop. 3. Wherefore the probability that any two independent events both happen is a ra- tio compounded of the probability of the i ft and the probability of the 2d. And in like manner confidering the ift and 2d event together as one event 5 the proba- bility that three independent events all happen is a ratio compounded of the probability that the two i ft both happen and the probability of the 3d. And thus you be accompanied with another to be determined at the fame time ? In this cafe, as one of the events is given, nothing can be due for the expectation of it ; and, confequently, thq value of an ex- peftation depending on the happening of both events muft be the fame with the value of an expedlation depending on the happen- ing of one of them. In other words ; the probability that, when one of two events happens, the other will, is the fame with the probability of this other. Call x then the probability of this other, and if - be the probability of the given event, and L ^ p b i> ^ the probability of both, becaufe 7; = - X .v* a* r: 4 r: the pro- *■ J si si b bability mentioned in thefe propofitions. may [ 383 ] may proceed If there be ever fo many fuch events j from whence the propofition is manifeft. Cor. I. If there be feveral independent events, the probability that the ift happens the 2d fails, the 3d fails and the 4th happens, &c. is a ratio compound- ed of the probability of the ift, and the probability of the failure of the 2d, and the probability of the failure of the 3d, and the probability of the 4th, &c. For the failure of an event may always be coniidered as the happening of its contrary. Cor. 2. If there be feveral independent events, and the probability of each one be ^, and that of its fail- ing be ^, the probability that the i ft happens and the 2d fails, and the 3d fails and the 4th happens, &c. will hQ abba^ &c. For, according to the algebraic way of notation, if ^denote any ratio and ^another, abba A^nott^ the ratio compounded of the ratios a^ b^ by a. This corollary therefore is only a particular cafe of the foregoing. Definition. If in confequence of certain data there arifes a probability that a certain event fhould happen, its happening pr failing, in confequence of thefe data, I call it's happening or failing in the I ft trial. And if the fame data be again re- repeated, the happening or failing of the event in confequence of thetn 1 call its happening or failing in the 2d trial j and fo on as often as the iame data are repeated. And hence it is manifeft that the hap- pening or failing of the fame event in fo many difFe- trials, is in reality the happening or failing of fo many diftind independent events exadiy fimilar to each other. Ddd 2 PROP. [ 384 ] K U Jr. y. If the probability of an event be a^ and that of its failure be b in each fingle trial, the probability of its happening p times, and failing q times xnp^q trials is E db^ if E be the coefficient of the term in which occurs d b* when the binomial a -|-^j ^"*"^ is ex- panded. For the happening or failing of an event in differ- ent trials are fo many independent events. Where- fore (by cor. 2. prop. 6.) the probability that the event happens the ift trial, fails the 2d and 3d, and hap- pens the 4th, fails the 5th, &c. (thus happening and failing till the number of times it happens be p and the number it fails be q) is abbab &cq. till the number of ^'s be^ and the number of Fs be ^, that is; 'tis d b\ In like manner if you confider the event as happening p times and failing q times in any other particular order, the probability for it is d b^\ but the number of difRrent orders according to which an event may happen or fail, io as in all to happen ^ times and fail q^ mp \ q trials is equal to the num- ber of permutations that aaaa bbb admit of when the number of d% is /, and the number of ^'s is q. And this number is equal to E, the coefficient of the term in which occurs d b^ when a^ ^^Hs ex- panded. The event therefore may happen / times and fail f in ^ -|- 5^ trials E different ways and no more, and its happening and failing thefe feveral dif- ferent ways are fo many inconfiflent events, the pro- bability for each of which is a^ b^^ and therefore by prop. [ 38s ] prop* I. the probability that fome way or other it happens p times and fails q times m p ^q trials is E a^ bK S E C T I O N 11. Poftulate. I. I Suppofe the fquare table or plane A B C D to be fo made and levelled, that if either of the balls o or W be thrown upon it, there fhall he the fame probability that it refts upon any one equal part of the plane as another, and that it muft ueceflarily reft fomewhere upon it. 2. I fuppofe that the ball W fliall be ift thrown, and through the point where it refts a line os fhall be drawn parallel to AD, and meeting CD and AB in J and <?; and that afterwards the ball O fliall be thrown / 4- ^ or ;^ times, and that its refting between AD and os after a fingle throw be called the hap- pening of the event M in a fingle trial* Thefe things fuppofed, Lem. I. The prQba-.n F » s H I K • L x\ bility that the point will fall between any two pints in the line A B is the ratio of the diftance between the two points to the whole line AB. Let any two points be named, as f and k in the line AB, and 8 through them parallel to A D draw fF^ b L meeting C D in F and L. Then if the red- angles Cj^ F 4 L A are com^ [ 386 ] commenfurable to each other, they may each be di- vided into the fame equal parts, which being done, and the ball W thrown, the probability it will reft fomewhere upon any number of thefe equal parts will be the fum of the probabilities it has to reft upon each one of them, becaufe its refting upon any differ- ent parts of the plane AC are fo many inconfiftent events ^ and this fum, becaufe the probability it Ihould reft upon any one equal part as another is the fame, is the probability it fliould reft upon any one equal part multiplied by the number of parts. Confequently, the probability there is that the ball W fhould reft fome- where upon F^ is the probability it has to reft upon one equal part multiplied by the numberof equal parts in Yh and the probability it rells fomewhere upon 6/ or LA, i.e. that it dont reft upon F^ (becaufe it muft reft fome- where upon A C) is the probability it refts upon one equal part multiplied by the number of equal parts in C/, L A taken together. Wherefore, the probability it refts upon Fi is to the probability it dont as the number of equal parts in F^ is to the number of equal parts in CJ] L A together, or as ^ b to Cf, LA together, or as/i to B/ A^ together. Where- fore the probability it reft upon F^ is to the proba- bility it dont as fb to B/, Kb together. And (com- ponendo imerfe) the probability it refts upon F^ is to the probability it refts upon ¥b added to the proba- bility it dont, as fb to A B, or as the ratio oi fb to AB to the ratio of AB to AB. But the probabi- lity of any event added to the probability of its failure is the ratio of equality; wherefore, the probability it reft upon F ^ is to the ratio of equality as the ratio of Jb to AB totheratioof AB to AB, or the ratio of equality j and therefore the probability it reft upon ¥b [ 387 3 F^ is the ratio of/3 to AB, But e^ h)fothefi ac- cording as the bail W falls upon F 3 or not the points will lie between /and b or not, and there- fore the probability the points will lie between/ and b is the ratio of /I to AB. Again s if the rectangles C/ F^ LA are not commenfurable, yet the laft mentioned probability can be neither greater nor lefs than the ratio of/3 to A B y for, if it be lefs, let it be the ratio of /^ to AB, and upon the line/3 take the points p and ty h that p t fhall be greater than fc^ and the three lines B^, pfy tK commenfurable (which it is evident may be always done by dividing A B into equal parts lels than half <r3, and taking p and / the neareft points of divifion to/and ^ that lie upon /3). Then becaufe Bp^ pt^ t A are commenfurable, fo arc the recSangles C/, D /, and that upon p f compleating the fquare AB. Wherefore, by what has been faid, the probability that the point will lie between p and if is the ratio of /if to AB. But if it lies between p and t it muft lie between / and b. Wherefore, the probability it fliould lie between / and b cannot be lefs than the ratio of p i to A B, and therefore muft be greater than the ratio of fc to AB (fince pt h greater than fc). And after the fame manner you may prove that the forementioned probability cannot be greater than the ratio of /3 to A B, it mull there- fore be the fame. Lem. 2. The ball W having been thrown, and the line os drawn, the probability of the event M in a fingle trial is the ratio of Ae? to AB. For, in the fame manner as in the foregoing lem- ma, .he probabili^ .bat .h= baU . being .h?cwf ftall refc [ 388 ] reft fomewhere upon D^ or between AD and so is is the ratio of A ^ to A B. But the refting of the ball between A D and s o after a fingle throw is the happening of the event M in a llngle trial. Wherefore the lemma is manifeft. PROP. 8. If upon BA you ere£t the figure BghikmK whofe property is this, that (the bafe B A being di- vided into any two parts, as A^, and 15 b and at the point of divifion b a perpendicular being ereded and terminated by the figure in m 5 and y^ x^ r repre- fenting refpe<5tively the ratio of bniy A A, and BA to A B, and E being the the coefficient of the term in which occurs a^ b'^ when the binomial aJ^tY'^^ is expanded) yz=:'Ex^ r ^. I fay that before the ball W is thrown, the probability the point fhould fall be- tween / and A, any two points named in the line AB, and withall that the event M fhould happen p times and fail q in p -^ q trials, is the ratio of fghikmby the part of the figure BghikmA in- tercepted between the perpendiculars y^, bm raifed upon the line A B, to C A the fquare upon A B. DEMONSTRATION. For If not 5 ift let it be the ratio of D a figure greater than fgbikmb to CA, and through the points Cy dj c draw perpendiculars to fb meeting the curve Kmigl^ in h,i,k*, the point d being fo placed that di (hall be the longeft of the perpendi- ^ culars [389] culars terminated by the line Jb^ and the curve AmtgB ; and the points e, d, c being fo many and fo placed that the reftangles, bk^ c /, e i, fb taken together fhall differ lefs from Jghikmb than D does y all which may be eafily done by the help of the equation of the curve, and the difference between D and the figure Jghikmb given. Then fince di is the longeft of the perpendicular ordinates that infift upon fbj the reft will gradually decreafe as they are farther and farther from it on each fide, as appears from the conftrudion of the figure, and confequently ^ ^ is greater than g/ or any other ordinate that in- fifts upon e/^. Now if Ao were equal to A^, then by lem. 2. the probability of the event M in a fingle trial would be the ratio of A ^ to A B, and confequently by cor. Prop. I. the probability of it's failure would be the ratio of B ^ to A B. Wherefore, if x and r be the two forementioned ratios refpeftively, by Prop. 7. the probability of the event M happening p times and failing q in p ^q trials would be Ex^ r^. But x and r being refpedlively the ratios of A ^ to A B and B^ to AB, if ;^ is the ratio of eh to AB, then, by conftrudtion of the figure A/B, y z=zEx^ r^. Wherefore, if A ^ were equal to A ^ the probability of the event M happening / times and failing q in p-\-q trials would be y^ or the ratio of eh to A B, And if Ao were equal to A/, or were any mean be- tween Ae and Af^ the laft mentioned probability for the fame reafons would be the ratio of y^ or fome other of the ordinates infifting upon ef^ to A B. But eh is the greateft of all the ordinates that infift upon ef. Wherefore, upon fuppofition the point fhould lie Vol. LIII. Eee any [ 390 ] any where between f and e^ the probability that the event M happens p times and fails qmp^q tri- als can't be greater than the ra,tio of ^^ to AB. There then being thefe two fubfcquent events, the ift that the point o will lie between rand/, the 2d that the event M will happen p times and fail q in p 4" f trials, and the probability of the ift (by lemma ift) is the ratio of ef to AB, and upon fup- pofition the Ift happens, by what has been now proved, the probability of the 2d cannot be greater than the ratio oi eh to A B, it evidently follows (from Prop. 3.) that the probability both together will hap- pen cannot be greater than the ratio compounded of that of f/ to AB and that of eh to AB, which compound ratio is the ratio oi fh to C A Where-^ fore, the probability that the point will lie between /' and ey and the event M happen p times and fail i is no. greater than the ratio of /A .o C A. And in like, manner the probability the point will lie be- tween e and ^, and the event M happen and fail as before, cannot be greater than the ratio of ei to C A. And pgain, the probability the point will lie betveeen d and Cy and the event M happen and fail as before, cannot be greater than the ratio of ^/ to C A. And laftly, the probability that the point will lie between c and by and the event M happen and fail as before, cannot be greater than the ratio of ^ ^ to C A. Add now all thefe feveral probabilities together, and their fum (by Prop, i . ) will be the probability that the point will lie fomewhere between f and b^ and the event M happen p times and fail q m p J^ q trials. Add likewife the correfpondent ratios together, and their fum will be the ratio of the fum of the antecedent's to [ 391 ] to their common confequent, i, e. the ratio of //&, eiy ci^ hk together to CA; which ratio is lefs than that of D to C A, becaufc D is greater than fhy e /, ci^ bk together. And therefore, the probability that the point o will lie between y* and 4 and withal that the event M will happen p times and fail q in p A- ^ trials, is lefs th^n the ratio of D to C A ; but it was fuppofed the fame which is abfurd. And in like manner, by infcribing reftanglcs within the figure, as eg^ dhy dk^ cm^ you may prove that the laft mentioned probability is greater than the ratio of any figure lefs than fghikmb to C A, Wherefore, that probability muft be the ratio of fghikmb to CA. Cor. Before the ball W is thrown the probability that the point o will lie fomewhere between A and B, or fomewhere upon the line A B, and withal that the event M will happen p times, and fail q in / 4" ? trials is the ratio of the whole figure A/B to C A. But it is certain that the point o will lie fomewhere upon A B. Wherefore, before the ball W is thrown the probability the event M will happen p times and fail q m p '\- q trials is the ratio of A / B to C A. PROP. 9. If before any thing is difcovered concerning the place of the point (?, it fhould appear that the event M had happened p times and failed qvixp^q trials, and from hence I guefs that the point lies between any two points in the line A B, as /and b^ and con- fequently that the probability of the event M in a An- gle trial was Ibmewhere between the ratio of A b to A B and that of A /to A B : the probability I am in E e e 2 the [ 392 ] the right Is the ratio of that part of the figure A /B defcribed as before which is intercepted between perpendiculars ereded upon A B at the points f and b^ to the whole figure A / B. For, there being thefe two fubfequent events, the firft that the point o will lie between f and b \ the fecond that the event M fhould happen / times and fail^y in p -|- q trials s and (by cor. prop. 8.) the original* probability of the fecond is the ratio of A / B to C A, and (by prop. 8.) the probability of both is the ratio oi f g h imb to C A 5 wherefore (by prop. 5) it being firft difcovered that the fecond has happened, and from hence I guefs that the firft has happened alfo, the probability I am in the right is the ratio of fghimb to A/B, the point which was to be proved. Cor. The fame things fuppofed, if I guefs that the probability of the event M lies fome where be- tween and the ratio of A ^ to A B, my chance to be in the right is the ratio oi K b m to A / B. Scholium- From the preceding propofition it is plain, that in the cafe of fuch an event as I there call M, from the number of times it happens and fails in a cer- tain number of trials, witiiout knowing any thing more concerning it, one may give a guefs where- abouts it's probability is, and, by the ufual methods computing the magnitudes of the areas there menti- oned, fee the chance that the guefs is right. And that the fame rule is the proper one to be ufed in the cafe of an event concerning the probability of which we [ 393 ] we abfolutely know nothing antecedently to any trials made concerning it, feems to appear from the follov/ing confideration ; viz. that concerning fuch an event I have no reafon to think that, in a certain number of trials, it fhould rather happen any one pofTible number of times than another. For, on this account, I may juftly reafon concerning it as if its probability had been at iirft unfixed, and then determined in fuch a manner as to give me no reafon to think that, in a certain number of trials, it fhould rather happen any one poflible number of times than another. But this is exadtly the cafe of the event M, For before the ball W is thrown, which determines it's probability in a fingle trial, (by cor. prop. 8.) the probability it has to happen p times and fail qinp '\- qox n trials is the ratio of A / B to C A, which ratio is the fame when p '\' q ox n\% given, whatever number ^ is ; as will appear by computing the magnitude of A / B by the method * of fluxions. And confequently before the place of the point o is difcovered or the number of times the event Mhas happened in n trials, I can have no reafon to think it ihould rather happen one pof-. fible number of times than another. In what follows therefore I Ihall take for granted that the rule given concerning the event M in prop. 9. is alfo the rule to be ufed in relation to any event concerning the probability of which nothing * It will be proved prefently in art. 4. by computing in the method here mentioned that A / B contrafted in the ratio of E to 1 is to C A as I to wf I xE : from whence it plainly follows that, antecedently to this contraction, A / B muft be to C A in the ratio of i to « + 1, which is a conftant ratio when n is given, whatever p is, at [ 394 ] ?it all Is known antecedently to any trials made or ob- ferved concerning it. And fuch an event 1 mall call an unknown event. Cor. Hence, by fuppofing the ordinates in the fi- gure A/B to be contradled in the ratio of E to one, which makes no alteration in the proportion of the parts of the figure intercepted between them, and applying what is faid of the event M to an unknown event, we have the following propofition, which gives the rules for finding the probability of an event from the number of times it adually h?ippens and fails, PROP, 10. If a figure be defcribed upon any bafe AH (Vid. Fig.) having for it's equation j)? m^^ r^ ; where )\ x^ r are relpecftively the ratios of an ordinate of the figure infifling on the bafe at right angles, of the fegment of the bafe intercepted between the ordinate and A the beginning of the bafe, and of the other fegment of the bafe lying between che ordinate and the point H, to the bafe as their common confequent, 1 fay then that if an unknown event has happened p times and failed q mp -^q trials, and in the bafe AH taking any two pomts as jT and t you eredt the ordinates fc^ tF at right angles with it, the chance that the probability of the event lies fomewhere be- tween the ratio of Af to A H and that of A ^ to A H, is the ratio of tFCf, that part of the before- defcribed figure which is intercepted between the two ordinates, to A C F H the whole figure infifting on the bafe AH. This is evident from prop. 9. and the remarks made in the foregoing fcholium and corollary. 5 Now [395 ] Now, In order to reduce the forego- ing rule to pradice, we muft iind the value of the area of the figure de- fcribed and die fe- veral parts of it fe- tt parated, by ordi- " nates perpendicu- lar to its bafe. For which purpofe, fuppofe A H rrr i and H O the fquare upon A H likewife ■=: i , and Cf will be z=y, and Af-=: x, and Hfz=: r, becaufe y^ ^ and r denote the ratios of Cy, A J] and II J^ refpedively to AH. And by the equation of the curve jy =Ar^r^and (be- caufe Af -^/H = A H) r -|- AT = I. Wherefore y z=zx^ X i-M ^ X' qx p -^ % qx q-i X X 9. X §'- 1 X q-'2. XX -\- &c. Now the abfcifle being X and the ordinate x the correlpondent area is x p*^ J (by prop. 10. caf. i. Quadrat. Newt.) ^ and the ordi- nate being g x the area is j' ^v y and iji like man- * Tis very evident here, without having recourfe to Sir Ifaac Newton, that the fluxion of the area AC/ being yxzzx^x — qx X + qxq^i X X &c. the fluent or area itfclf is x^ -. q ><>__ + q X £2? ^ ^^ ^ &C. ner ner of the reft. Wherefore, the abfciiTe being x and P P+i the ordinate J or ^ ^gx -^ &cc. the correfpondent ^ + 1 p + 2 ^4-3 area is x ^gxx -\- q K g-t X x - ^ x q-i X P^i p+2 2 p-h3 2 ^-2 X ^ + &c* Wherefore, if x == Ay' = A^, 3 P + i- A.H andjy = C/=C^ then ACfz= ACf =zx AH HO >+i ^+2 /' + 3 -^q y. X ■^-qxg-ix X — &c. ^^^,^,g,g„p^0 MBwnpWHHH* 11,1.1 M l > m '»mtmmm* m From which equation, if ^^ be a fmall number, it is eafy to find the value of the ratio of A C/ to HO. and in like manner as that was found out, it will ap- q+ I pear that the ratio of HCf to HO is r — ^ x q+l q+t ?+3 f+4 f+2 2 f+3 ^ 3 ^ + 4 which feries will confift of few terms and therefore is to be ufcd when p is fmalL 2. The fame things fuppofed as before, the ratio of ^+1 ^+2 AC/toHOis^ r^^A-qy.x r^ +?X *^ •Hi«HaiHMk-w« ' ' ■ -I* ■Ill |ii1 1 I «■ ■■■ «i I ■ ^+1 J»+I ^+2 ^+1 j? + 3 ^+4 g^i: x ^ r^ ^ + y X j^-i X y-a x ^ ^^''^ + JTT F+3 f+i /»+^ /^+3 ^+4- &c. [ 397 ] ice. J^x X 9' X J--! X &c. X t where « =s ^gl0mmmmmfi» mmmmmrmmtm iMMMwriHMHMMiw nrntH^ n+i ^+1 p-jrZ » f + t p J^q. For this feries is the fame with :>c ^•— q X p+2 p+i X &c. fet down in Art. ift. as the value of the ■M p'jr2 ratio of A Cf to H O j as will eafily be fern by put- ting in the former inftead of r its value i-x, and expanding the terms and ordering them according to the powers of x. Or, more readily, by comparing the fluxions of the two feries, and in the former inftead of r fubftituting ^x^. * The fluxion of the firft feries i$ pc r x + gx t^ r + P-^^q^t P + ^a^M P + ^q^2 qx r X + q X q-'l X x r r- + q X f-*I XX r j^ jf> + ' /+^ i^+* ^+* ^Hh* + q X f-i X f-3 X X r^**? &c. or, fubftituting - i for r^ X r X '-^ a X r^ x + qx r^ ^— fX y— i x p+% p+% ^^ ^ r^ "^x + q X q^t X X r^^ x &c. which, as all the mmmmmmmm P + 2 p + I p + 2 terms after the firft deftroy one another, is eq ual to x^ r^ x r=, x^ X i— %y|^ ;v = ^i^ ;c X I -^qx-^-qX q — l ^"^ &c. == A'^ ^ — ^+1 p + 2 2 t /• • f y ;^ + f X f-i *• ^ &c, =: the fluxion of the latter fcnes P+ 1 * p + S3t or of ^ — f X ^ &c* The two feries therefore arc P+I P+2 the fame. Vol. LIIL Fff 3* ^^ [ 39B ] 3- In like manner, the ratio of HCf to HO i« r x^ -\- p X r x^'^ +/^ X p-^i X ^ ^___ ^^ &c. 4. If E be the coefficient of that term of the bi- nomical a -}- ^[/' + ^ expanded in which occurs ^/^ ^^> the ratio of the whole figure ACFH to HO is TTi ^ "e' ^ being =/) -f-j^. For, when Ayi=: A H x^= i^ r=io. Wherefore, all the terms of the fe- lies fet down in Art. 2. as expreffing the ratio of A Cf to H O will vanifh except the laft, and that becomes -tt X ;tt X S X &c. X J . But E being the coefficient of that term in the binomial a -\- b^ expanded in which occurs a^ b^ is equal to t±i X ^i^ X &c. X ^. And,becaufe Ay isfup- pofed to become = AH, A C/= A C H. From whence this article is plain. 5, The ratio of AC/ to the whole figure ACFH 'mmm^mmi'mmi^nmmmitm is (by Art. i. and 4.) ;^ -|- i x E x ^ — J' X X 4" y >^ ?-^ X ^ ^^' ^^^ ^f:) ^^ ^ expreffes >+2" '^T" A+ 3 the ratio of A/ to AH, X fhould expreis the ratia of A^ to AHi the ratio of AF/ to ACFH would be «-^i X E X X — ^X +$'X?-i X X — &c. and confequ&ntly the ratio of /FC/ to ACFH is ;^-j- 1 X E X into the difference between [ 399 ] betwetn the two ferks. Compares this with jprop, io# and we ihall have the following pradical mlc. RTT T 17 T \J Li ill I. If nothing is known concerning an event but that it has happened ptimts and failed qinpJ^govn trials, and from hence I guefs that the probability of its happening in a fingle trial lies fomewhere between any two degrees of probability as X and ;.,jhe chance I am in the right in my guefs is n'\^i X Ex into the difference between the feriesX^''"^ p+2 p + 3 p+i 4-f X ?-r X X — &c. and the MlMaali^ mmmmm0mffm. P+i p + ^ P+3 feriep x -^ f ^ +f X f-i X x — &c. E ^+1 p + 2 2 j> + 3 being the coefficient of a^ b^ when ^-|- bX is expanded. This is the proper rule to be ufed when j' is a fmall number ; but if q is large and p fmall, change every where in the feries here fet down p into q and q into/ and X into r or i-at, and X into R =: i-Xj which will not make any alteration in the difference between the two feriefes. Thus far Mr. Bayes*s effay. With refped to the rule here given, it is further to be obferved, that when both / and q are very large numbers, it will not be fKjflible to apply it to practice on account of the multitade of terms which the fe- riefes in it will contain. Mr. Bayes, therefore, by F f f 2 an [ 400 ] an Inveftigation which it would be too tedious to give here, has deduced from this rule another, which is as follows. RULE £• If nothing is known concerning an event but that it has happened p times and failed y in ^ + q or n trials, and from hence I guefs that the probabihty of its happening in a fingle trial lies between ^ 4- 2J and «k mm t — 2; J if /»* ■=— a = *, b=i 1, E the coefficient » pq n n of the term in which occurs a^ h^ when a -^A] is expanded, and 2 = ^^ — - X —;==:=• X E 12^ i^ x by the fenes mz -— 1 X -— - — -^ ^ - ^^ X 1 X — ^ X — X — &c. my chance to be in the right is greater than ■■ 2 X ^ I 4- 2t E ^^ ^s' ^- 2 E ^^ ^^ * and lefs than 2 s « 1-2 E ii/^f — 2 E ^i^ if. And if / = f my chance is 2 2 exadly. « * In Mr. Bayes's manufcript this chance is made to be gieater ,han — .i^^ and lefs than ?#^^. The third term in the two dlvifors, as I have given them, being omitted. But this being evidently owing to a fmall overfight in the dedui3:ipn of this rule, which I have reafon to think Mr. Bayes had himfelf difcovered, I have ventured to correal his copy^ and to give the Jule as I am fatisficd it ought to be given. In [ 401 ] In order to render this rule fit for ufe In all cafes it is only neceflary to know how to find within fuffi- cient nearnefs the value o£ E af h and alfo of the m^ %' feries m z &c ^. With refpeffc to the former Mr, Bayes has proved that, fuppofing K to fignify the ratio of the quadrantal arc to it's radius, E a^ if will be equal to ^;-y==— r x by the ratio whofe hyperbo-- ^ 12 n /* ? 300 « p ' ' ' " I — I I I I I I mm — ■———It Jt , I II I I r I £^ > 1260 ^ ^ p" p 1680 ^ «» ""p ■" T + 7-00 X -7 — TT •— - -r &c. where the nume- ral coefficients may be found in the following man- ner. Call them A, B, C, D, E, &c. Then A r=r 2. 2. 3 3.4 2* 4- 5 3 2. 6* 7 loB + A jQ I ^^ 3sC+2iB+A g _ I 5 * 2.8.9 7 * 2. 10 .11 126 C + 84D -F 36 B + A p 1^ 9 2. 12. 13 ""^ * A very few terms of this feries will generally give the hyper- bolic logarithm to a fuiEcient degree of exaSnefs. A fimilar fe- ries has been given by Mr. De Moivre, Mr. Simpfon and other eminent mathematicians in an exprefEon for the fum of th^ lo* garithms of the numbers 1, 2, 3, 4, 5 to x^ which fum they have affer ted to be equal to | log. r + a* 4- | x log» x — x + -TTx — -i^x + "ttVo/ &c. c denoting the circumference of a circle whofe radius is unity. But Mr. Bayes^ in a preceding pa- per in this volume, has demonftrated that, though this expreffion will very nearly approach to the value of this fum when o»ly a proper number of the firft terms is taken, the whole feries cannot exprefs any quantity at all, becaufe, let x be what it will, there will be always a part of the feries where it will begin to diverge. This obfervation, though it does not much afFe£l the ufe of this feries, feems well worth the noticeof mathematicians, 462 t 4*^2 ] ^^.±J12£±l^jA±Il±tJt &c. where the co- efficients of B, C, D, E, F, &c. in the values of D, E, F, &c. are the 2, 3, 4, &c. higheft coeffid- cnts in a +^| > ^ + ^I j a -^ ^\ \ &c. expanded; affixing in every particular value the leaft of thefe eoefficents to B, the next in magnitude to the fur- theft letter from B, the next to C, the next to the furtheft but one, the next to D, the next to the fur- theft but two, and £0 on *• With refpeft to the value of the feries mz — f-L 4. 2li )^ ^IS. &c. he has obferved that it may be calculated direcftly when mz is lefs than i, or even not greater than VJ: but when m z is much larger it becomes imprafticable to do this j in which cafe he fhews a way of eafily finding two values of it very nearly equal between which it's true value muft lie. The theorem h? gives for this purpofe is as fol- lows. Let K, as before, ftand for the ratio of the qua- drantal arc to its radius, and H for the ratio whofe hyperbolic logarithm is ^ ~ |l^ + ^^ ~ ^xF~ &c. Then the feries ^^ — —^&c, will be greater or lefs than the feries '-1— x «-=--—--— x .1^ « 1 ^ - . I— « zmz «+2 « -j- 4 X 4 w^ 2^ ^ This method of finding thefe coefficients I have deduced from the demonftratign of the third lemma at the end of Mr. Simpfon's Trcatife on the Nature and Laws of Chance. .2 W2* Z* 3«' [ 403 ] T + 3 TZZ2^p\% -]- 4, 2^5 x^'* _____ n X ^ - ^ ^ . . J ., ; --X — &c. continued to any number of terms, accord- ing as the laft term has a pofitive or a negative fign before it. From fubftituting thefe values of Ea^ if and m z m^ z^ , n — 2 m^ z^ 4- -. — X &c. in the 2d rule arifes a 3 * 2w 5 3d rulq, which is the rule to be ufed when mz is of fome confiderabie magnitude. RULE 3. If nothing is known of an event but that it has happened f> times and failed q in ^ -f y or n trials^ and from hence I judge that the probability of it's happening in a fingle trial lies between - -{- z and ^ -~ z my chance to be right is greater than mmmmmmMmtmm'90^ X I zm" z" n ? + I and lefs than J^M^^ .^ 2 ZwKpq-hn^i-hn "^ multiplied by the 3 terms 2 H — ~4 x ^^ ?+2 «+i 1 1 '-2 m'' z"^ 5v '■» <» /^ where nf'y K, ^ and H ftand for the quantities already explained. An [ 404 ] An APPENDIX CONTAINING An Application of the foregoing Rules to fome parti- cular Cafes. np H E firft rule gives a diredt and perfed Iblution -■• in all cafes ; and the two following rules are only particular methods of approximating to the fo- lution given in the firft rule, when the labour of ap- plying it becomes too great. The firft rule may be ufed in all cafes where either p ov q are nothing or not large. The fecond rule may be ufed in all cafes where mz is lefs than VY; and the 3d in all cafes where m" z" is greater than I and lefs than - , if « is an even number and very large. If n is not large this laft rule cannot be much wanted, becaufe, m decreafing continually as n is diminifhed, the value of z may in this cafe be taken large, (and therefore a confidcrable interval had be- tween Z *- 2J and ±L 4- z^) and yet the operation be n n ' carried on by the 2d rulej ox mz not exceed x/y. But in order to (hew diftinftly and fully the nature of the prefent problem, and how far Mr. Bayes has carried the folution of it ; I fliall give the refult of this folution in a few cafes, beginning with the loweft and moft fimple. Let [ 405 ] Let us then firft fuppofe, of fuch an event as that called M in the eflay, or an event about the proba- bility of which, antecedently to trials, we know no- thing, that it has happened onccy and that it is en- quired what conclufion we may draw from hence with refped: to the probability of it's happening on a fecond trial. The anfwer is that there would be an odds of three to one for fomewhat more than an even chance that it would happen on a fecond trial. For in this cafe, and in all others where q is H*«PltM««aM««MMl • pj^l pj^l nothing, the expreflion n\^i %1L —a; or X * — a:^ "^ gives the folution, as will appear from confidering the firft rule. Put therefore m this expreflion J+7 = 2, X = i and a; = 4. and it will be I —T|* or 1.5 which fhews the chaoce there is that the probability of an event that has happened once lies fomewhere between i and 4. ; or (which is the fame) tlie odds that it is fomewhat more than an even chance that it will happen on a fecond trial *. In the fame manner it will appear that if the event has happened twice, the odds now mentioned will be feven to one ; if thrice, fifteen to one 5 and in gene- ral, if the event has happened p times, there will be an odds of 2/^ + » — i to one, for more than an equal chance that it will happen on further trials. Again, fuppofe all I know of an event to be that it has happened ten times without failing, and the * There c^Hj, I fuppofe, be no reafon for obferving that on this ful*je<ft unity is always made to fland for certainty, and -• for an isven chance. Vol. LIII. G g g enquiry [ 4.06 ] enquiry to be what reafon we fliall have to think we are right if we guefs that the probability of it's hap- pening in a fingle trial lies fomewhere between -i.^ and -*, or that the ratio of the caufes of it's happen- ing to thofe of it's failure is fome ratio between that of fixteen to one and two to one. ^. , Here/-|- i =z=: ii, X==4|. and x=z^ and X — • x^+^ =411" — H" == .501-2 &c. The anfwer therefore is, that we (hall have very nearly an equal chance for being right. In this manner we may determine in any cafe what conclufion we ought to draw from a given number of experiments which are unoppofed by contrary experiments. Every one fees in general that there is reafon to expecfl an event with more orlefs confidence according to the greater or lefs number of times in which, under given circumftances, it has happened without failing ; but we here fee exaftly what this reafon is, on what principles it is founded, and how we ought to regulate our expedations. But it will be proper to dwell longer on this head» Suppofe a folid or die of whofe number of fides and conftitution we know nothing ; and that we are to judge of thefe from experiments made in throwing it In this cafe, it fhould be obferved, that it would be in the higheft degree improbable that the folid fhould, in the firfl trial, turn any one fide v^hich could be afTigned before hand ; becaufe it would be known that fome fide it mufl turn, and that there was an in- finity of other fides, or fides otherv^ife marked, which it was equally likely that it fhould turn. The firfl A throw I 407 ] throw only (hews that it has the fide then thrown, without giving any reafon to think that it has it any one number of times rather than any other. It wilJ appear, therefore, that after the firft throw and not before, we fhould be in the circumftances required by the conditions of the prefent problem, and that the whole efFed of this throw would be to bring us into thefe circumftances. That is : the turning the fide firft thrown in any fubfequent fingle trial would be an event about the probability or improba- bility of which we could form no judgment, and of which we fhould know no more than that it lay fomewhere between nothing and certainty. With the fecond trial then our calculations muft begin; and if in that trial the fuppofed ibiid turns again the fame iide, there will arife the probability of three to one that it has more of that fort of fides than of all others ^ or (which comes to the fame) that there is fomewhat in its conftitution difpofing it to turn that fide ofteneft : And this probability will increafe, in the manner already explained, with the number of times in which that fide has been thrown without failing. It fhould not, however, be imagined that any number of fuch experiments can give fufficient reafon for thinking that it would never turn any other fide. For, fuppofe it has turned the fame fide in every trial a million of times. In thefe circumftances there would be an improbability that it had lefs than 1.400,000 more of thefe fides than all others; but there would alfo be an improbability that it had above 1.600,000 times more. The chance for the latter is expreffed by 4.4.^^4. raifed to the millioneth power fubftraded from unity, which is equal to .4647 &c.and G g g 2 the [ 408 ] the chance for the former is equal to i4^4^-o4i rzKci to the fame power, or to .4895; which, being both lefs than an equal chance, proves what I have faid. But though it would be thus improbable that it had ahve 1.600,000 times more or kjs than 1400,000 liines more of thefe fides than of all others, it by no means follows that we have any reafon for judging that the true proportion in this cafe lies fomewhere between that of 1,600,000 to one and 1400,000 to one. For he that will take the pains to make the calcula- tion will find that there is nearly the probability ex- preflfed by .527, or but little more than an equal chance, that it lies fomewhere between that of 600,000 to one and three millions to one- It may deferve to be added, that it is more probable that this proportion lies fomewhere between that of 900,000 to I and 1.900,000 to i than between any other two proportions whofe antecedents are to one another as 900,000 to 1.900,000, and confequents unity. I have made thefe obfervations chiefly becaufe they arc all ftridly applicable to the events and appear- ances of nature. Antecedently to all experience, it would be improbable as infinite to one, that any par- ticular event, beforehand imagined, fhould follow the application of anyone natural objed to another ^ becaufe there would be an equal chance for any one of an infinity of other events. But if we had once feeii any particular effed:s, as the burning: of wood on putting it into fire,, ol the falling of a^ftone on de- taching it from all contiguous objefts , then the con^- clufions to be drawn from any number of fubfequent events of the fame kind would be to be determined ill the f^me manner with the eonclufions juft mcn^ tioned relating to the conftitution of the folid I have fuppofed [ 409 ] fuppofed. In other words. The firft experi- ment fuppofed to be ever made on any natural obje<5t would only inform us of one event that may follow a particular change in the circumftances of thofe objedts ; 3ut it would not fuggeft to us any ideas of uniformity in nature, or give us the leaft reafon to apprehend that it was, in that inftance or in any other, regular ra- ther than irregular in its operations. But if the fame ey^nt has followed without interruption in any one or more fubfequent experiments, then fbme degree of uniformity will be obferved ; reafon will be given to exped the fame fuccefs in further experiments, and the calculations direded by the folution of this pro- blem may be made, ; ^ ^ One example here it will not be amrfs to give. Let us imagine to ourfelves the cafe of a perfon juft brought forth into this, world and left to colled from his obfervation of the order and courfe of events what powers and caufes take place in it. The Sun would, probably, be the firft objed that would engage his atten- tion; but after lofing it the firft night he would be en- tirelyignoran t whether he fliould ever fee it again. He would therefore be in the condtion of a perfon making a firft experiment about an event entirely unknown to him. But let him fee a fccond appearance or one return of the Sun, and an expedation would be raifed in him of a fecond return, and he might know that there was an odds of 3 to i ioxfome probability of this. This odds would increafe, as before reprefented, with the number of returns to which he was witnefs. But no finite number of returns would be fufficient to produce abfolute or phyfical certainty. For let it he fuppofed that he has feen it return at regular and ftated intervals a million of times. The conclufions 5 this [ 410 ] this would warrant would be fuch as follow ' '■■"' ■ ■ ■ ^ There would be the odds of the millioneth power of 2y to one, that it was likely that it would return again at the end of the ufual interval. There would be the probability expreffed by .5352, that the odds for this was not ^r^^^^r than i. 600,000 to i ; And the pro- bability expreffed by .5105, that it was not kfs than 1.400,000 to I. It fhould be carefully remembered that thefe de- dudions fuppofe a previous total ignorance of nature. After having obferved for fome time the courie of events it would be found that the operations of nature are in general regular, and that the powers and laws which prevail in it are ftable and parmanent. The confideration of this will caufe one or a few experi- ments often to produce a much ftronger expedation of fuccefs in further experiments than would otherwife have been reafonable j juft as the frequent obfervation that things of a fort are difpofed together in any place would lead us to conclude, upon difcovering there any objed of a particular fort, that there are laid up with it many others of the fame fort. It is obvious that this, fo far from contradiding the foregoing de- dudions, is only one particular cafe to which they are to be applied. What has been faid feems fufEcient to Ihew us what concluiions to draw from uniform experience. It demonftrates, particularly, that inftead of proving that events will always happen agreeably to it, there will be always reafon againft this conclufion. In other words, where the courfe of nature has been the moft conftant, we can have only reafon to reckon upon a recurrency of events proportioned to the degree of this [ 411 ] this conftancyj but we can have no reafon for thin Sl- ing that there are no caufes in nature which will ever inrerfere with the operations of the caufes from which this conftancy is derived, or no circumftances of the world in which it will fail. And if this is true> fup- poling our only data derived from experience, we fhall find additional reafon for thinking thus if we ap- ply other principles, or have recourfe to fuch conifi- derations as reafon, independently of experience, can fuggeft. But I have gone further than I intended here ^ and it is time to turn our thoughts to another branch of this fubje£t: I mean, to cafes where an experiment has fometimes fucceeded and fometimes failed. Here, again, in order to be as plain and explicit as poflible, it will be proper to put the following cafe, which is the eafieft and fimpleft I can think o£ Let us then imagine a perfon prefent at the drawing: of a lottery, who fncwsUin'g of its fchctne or of the proportion of Blanks to Prizes in it. Let it further be fuppofed, that he is obliged to infer this from the number of blanks he hears drawn compared with the number of prizes ^^ and that it is enquired what con- clufions in thefe circumftances he may reafonably make. Let him firft htzt ten blanks drawn and one^nzc^ and let it be enquired what chance he will have for be- ing right if he guefles that the proportion of Hanh to prizes in the lottery lies fome where between the pro- portions of 9 to I and 1 1 to i . ^ Here taking X = ^4. x= A./=ro, q= 1, n^i i, E=; II, the required chance, according to the firft: rule. [ 412 ] rule, is tt+ I X E into the difference between — _ />-f I 9^ P + 2 and /»+! p+2 X p+i qx 12 X II ^^+2 ^+1 />+2 rrv' III" JO » — . 121 12 II 1. 10 ti 1 lO 12. .07699 12 II 12 &c. There would therefore be an odds of about 923 1076, or nearly 12 to i againfl his being right. Had he gueffed only in general that there were lefs than 9 blanks to a prize, there would have been a proba- bility of his being right equal to .6589, or the odds of 65 to 34. Again, fuppofe that he has heard 20 blanh drawn and z prizes y what chance will he have for being right if he makes the fame guefs ? Here X and x being the fame, we have «= 22, fz=.2o^ f = ^j £ = 231, and the required chance "^^-fl p + 2 /»+3 equal to «i+i x E x X - j'X + j'XJ'-i xX 'p + 1 p + 2 P + 3 p+i X P + 2 p+3 qx -j-^X^'-lX^ = .10843 &c. ^+1 p + % 2 ^ + 3 He will, therefore, have a better chance for being right than in the former inftance, the odds againft him now being 892 to 108 or about 9 to i. But Ihould he only guefs in general, as before, that there were lefs than 9 blanks to a prize, his chance for be- ing right will be worfe ; for inftead of .6589 or an odds of near two to one^ it will be .584, or an odds of 584 to 415. Suppofe, [ 413 ] Suppofe, further, that he has heard 40 bknh drawn and 4 prizes \ what will the before -mention- ed chances be ? The anfwer here is .1525, for the former of thefe chances; and .527, for the latter. There will, there- fore, now be an odds of only 54. to i againft the proportion of blanks to prizes lying between 9 to i and II to I ; and but little more than an equal chance that it is lefs than 9 to i. Once more. Suppofe he has heard 100 blanh drawn and i o prizes. The anfwer here may flill be found by the firft mle; and the chance for a proportion of blanks to prizes /g/S than 9 to i will be .44109, and for a pro- portion jr^^/^r than II to I .3082. It would there- fore be likely that there were not fewer than 9 or more than 1 1 blanks to a prize. But at the fame time it will remain unlikely * that the true proportion fhould lie between 9 to i and 1 1 to i, the chance for this being .2506 &c. There will therefore be jftill an odds of near 3 to i againft this- From thefe calculations it appears that, in the clr- cumftances I have fuppofed, the chance for beinff right in gueffing the proportion of blanks to prizes to be nearly the fame with that of the number of blanks * I fuppofe no attentive perfan will find any difficulty In this. It is only faying that, fuppofing the interval between nothing and certainty divided into a hundred equal chances, there will be 44 of them for a lefs proportion of blanks to prizes than 9 to i, 31 for a greater than 11 to i, and 25 for fome proportion be- tween 9 to I and II to 1 5 in which it is obvious that, though one of thefe fuppofitions muft be true, yet, having each of them more chances againft them than for them, they arc all feparately unlikely. Vol. LIII. H h h drawn [ 414 ] drawn in a given time to the number of prizes drawn, is continually increafing as thefe numbers increafe j and that therefore, when they are confiderably large, this conciufion may be looked upon as morally cer- tain. By parity of reafon, it follows univerfally, with refpeft to every event about which a great number of experiments has been made, that the caufes of its happening bear the fame proportion to the caufes of its failing, with the number of happenings to the number of failures; and that, if an event whofe caufes are fuppofed to be known, happens oftener or feldomer than is agreeable to this conciufion, there will be reafon to believe that there are fome unknown caufes which difturb the operations of the known ones. With rcfpedl, therefore, particularly to the courfe of events in nature, it appears, that there is demonftrative evidence to prove that they are derived from permanent caufes, or laws originally eftabliftied in the conftitution of nature in order to produce that order of events which we obferve, and not from any of the powers of chance *. This is juft as evident as it would be, in the cafe I have infifted on, that the reafon of drawing lo times more blanks th^n prizes in millions of trials, was, that there were in the wheel about fo many more blanks than prizes. But to proceed a little further in the demonftration of this point. We have feen that fuppofing a perfon, ignorant of the whole fcheme of a lottery, (hould be led to con- jecture, from hearing loo blamsaad lo prizes drawn^ * See Mr, De Moivre's Dodrine of Chances, pag. 250. that [ 415 3 that the proportion of blanks to prizes in the lottery was fbme where between 9 iio i and xi to 1/ the chance for his being right would be ♦2506 &c. Let now enquire what this chance would be in fome h^her cafes. Let it be fuppofed that blanks haye been drjiwn 1000 times, and prizes 106 times In iroo trials. In this cafe the powers of 3£ gnd x rife fo high, and the number of terms in the two ferides X ^ qX &c« and a; — §^x &c. become to obtain the anfwer by the rf rule. Tb ncceffiiy, therefbrc, to ba^ rccourfe to thefecond rule. But in order to make ufe of it, the inmryal Tbetween X and ^ muft be a little altered. 44 *- -^ is ^4.^ audi rilerefbre the interval between 4^"^ - -^^ and 44 TT*3r will be nearly the fame with the interval be-* tweeh 4w ^^ iiy oi^ly fomewhat larger. If then wc make the queftion to be ; what chance there wbiild be (fiippofinsr no more known thm that blgnka pe been dSwn^ooo times and prizes lao tlmm in iioo trials) that the probability of drawing a blank in a fingle trial would lie fomewhere between ^^.-^^4:^ and 44 4" TTTir w^ ^^^^ have a queftion of thfe lame kind with the preceding queftiotfiSi and deviate but little from the limits dBgned in fiiembi The aniwer, according to the fccond eul^ m &^ litis chance is greater than i^ z Ea^ &^ + zl^ a?Tf n Hhh 2 and [ 4i6 ] a X W» n i l »i mmmmmmmm ■ ■ » i m j i m — — i— iM and lefs than 1--2 E ^^^ d^ ^-a E^^ ^f, E being n-^ 1 n n .3 ~3 ^ .JtLXxBa'^ p^ Xmz^ + — f x — -- &c, V « ^ 3 2» 5 By making here 1000 z=zp iooz=:q iioo=:« being the ratio whofe hyperbolic logarithm is -^'^ X I I I II I I.I I I I tf«.^ n p q 360^ n^ p f 1260 «^ ^^ f^ and K the ratio of the quadrantal arc to radius 5 the former of thefe expreffions will be found to be .7953, and the latter .9405 &c. The chance enquired after, therefore, is greater than .7953, and lefs than .9405* That is; there will be an odds for being right in gueff- ing that the proportion of blanks to prizes lies marly between 9 to i and 1 1 to i, (or exa^Iy between 9 to I and 1 1 1 1 to 99) which is greater than 4 to i, and lefs than 16 to i. Suppofe, again, that no more is known than that Hanks have been drawn 10,000 times and prizes 1000 times in iiooo trials ^ what will the chance now mentioned be? Here the fecond as well as the firft rule becomes ufelefs, the value of mz being fo great as to r ender it fcarcely poffible to calculate diredly the feries Hi^ ^^+2l!x^^&c. The third rule, therefore, 3 3t» 5 muft be Ujfed ; and the information it gives us is, that the required chance is greater than •97421, or more than an odds of 40 to i. By [ 4^7 ] By calculations fimilar to thefe may be determined univerfally, what expedtations are warranted by any experiments, according to the different number of times in which they have fucceeded and failed; or what fhould be thought of the probability that any particular caufe in nature, with which we have any acquaintance, will or will not, in anyfingle trial, produce an efFed that has been conjoined with it. Moft perfons, probably, might expert that the chances in the fpecimen I have given would have been greater than I have found them. But this only fhews how liable we are to error when we judge on this fubjedt independently of calculation. One thing, however, fhould be remembered here; and that is, the narrownefs of the interval between ^% and 44, or between 4.4 + ^4^ and 4.^ — ^4.^. Had this interval been taken a little larger, there would have been a confiderable difference in the ref^Us of the calculations. Thus had it been taken double, or z =:^V> ^' would have been found in the fqurth in- ftance that inftead of odds againft there were oddi for being right in judging that the probability of draw- ing a blank in a lingle trial lies between 44 -|- ^ and Xr TT* The foregoing calculations further fhew us the ufes^ and delfts of the rules laid down in the effay. *Tis evident that the two laft rules do not give us the required chances within fuch narrow limits as could be wiihed; But here again it fhould be confi- dered, that thefe limits become narrower aqd narrow- er as §^ is taken larger in reipeft of py arid when^ and y are equal, the exadl folution is given it% all cafes by the fecond rule. Thefe two rules therefore afford a direction I ^,1 8 ] tDc^f|ti%t4fent that may be of tehfideiv llbte^jie^Hij&mi ijsrleift {liall difcover a better ajji ^MAstinti 3iC©Ttfig> vdie of the two feries's in the 5{fi^|jlilvJ|g|iiirt^(p|ftoIirecommends the ibltition iil \hh m^ 1^ A#l¥t H^'compleat in thofe cafes whePi Mstrflliisli p^j mbfttcwaiited, and where Mr. De Moi^^#it)ltobifofc lise inverfe problem can give lilllet«s|J rf^3#f ^<^^^I«I mean, in all cafes where ei- tM4 pi0|o^i<a«eco|i|ai cohfiderable magnitude. Iwi ^iMSfft ^fes,8idl ts«Henboth / and ^ are very Confider- ab%,«ft igpotf diffictttd to perceive the truth of whafc U^^^tem^e 4mmkMmbd, or that thefe is reafoii tti feillevl4a geffeidl te?#ife chances for the happening y%^5,e\^ftlWs:yj«h#Hdhances for its failOre in the feife ratjb w'i^h t^it b£|& to q. But we fhall be greatly l3beei?ed#i3Wd|i«ige:ihlijis manner when either /bf' IfMfM^^ii AhdaiiscftJiH' fuch cafes the D^jifa are nol ^fBtlrfflpfoftifeSverfie^ekaa: probability of ian events yet lil^J(^r|lgfleelible3io be? able to find the limits be* tyeen3^(M;h:iitfe fei^abfe to think it muft lie, and aVi tb le I^feWdeietriline the precife degree of aflent kM^,Mmmmp<hrm^xiMms or aflertions relating to them. •Xf^Sirii^teyirW^tfekk have found out a methocl of confi- iteril)lgiii|)Sb«i%?»fena|^bxiniation in the ad and ad rules by diSS#Pl%ill»f rtf eg^feFi^on 1+2 E tf^ ^y + 2 £ i7^ ^ comes ffii||Ktt#f rM^ii i¥(^li&rbe given. LIIL An